Blue Origin LLC, the space-transportation company run by Jeff Bezos, has won a contract to provide engines for a potential rival’s next-generation rocket, further roiling a tumultuous industry and vaulting Mr. Bezos into the lucrative market for Pentagon satellite launches.

United Launch Alliance LLC—a joint venture between Boeing Co. and Lockheed Martin Corp. that launches U.S. military and spy satellites into orbit—on Thursday announced that after a lengthy competition, it picked Blue Origin’s BE-4 engine for its Vulcan...